EPM RT HYBRID FINITE ELEMENT METHOD APPLIED TO THE ANALYSIS OF FREE VIBRATION OF SPHERICAL SHELL

Transcription

1 EPM T 03-0 HYBID FINITE ELEMENT METHOD APPLIED TO THE ANALYSIS OF FEE VIBATION OF SPHEICAL SHELL M. Meaa, A.A. Laks Départemet de Gée mécaque École Polytechque de Motréal Avrl 03

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3 EPM-T-03-0 HYBID FINITE ELEMENT METHOD APPLIED TO THE ANALYSIS OF FEE VIBATION OF SPHEICAL SHELL M.Meaa, A.A.Laks Départemet de gée mécaque École Polytechque de Motréal Avrl-03

4 03 Mohamed Meaa, Aou A. Laks Tous drots réservés Dépôt légal : Bblothèque atoale du Québec, 03 Bblothèque atoale du Caada, 03 EPM-T-03-0 Hybrd fte elemet method appled to aalyss of free vbrato of sphercal shell par : Mohamed Meaa, Aou A. Laks Départemet gée mécaque École Polytechque de Motréal Toute reproducto de ce documet à des fs d'étude persoelle ou de recherche est autorsée à la codto que la ctato c-dessus y sot metoée. Tout autre usage dot fare l'objet d'ue autorsato écrte des auteurs. Les demades peuvet être adressées drectemet aux auteurs (cosulter le bott sur le ste ou par l'etremse de la Bblothèque : École Polytechque de Motréal Bblothèque Servce de fourture de documets Case postale 6079, Succursale «Cetre-Vlle» Motréal (Québec Caada H3C 3A7 Téléphoe : ( Télécope : ( Courrer électroque : bblo.sfd@courrel.polymtl.ca Ce rapport techque peut-être repéré par auteur et par ttre das le catalogue de la Bblothèque :

5 Abstract I ths study, free vbrato aalyss of sphercal shell s carred out. The structural model s based o a combato of th shell theory ad the classcal fte elemet method. Free vbrato equatos usg the hybrd fte elemet formulato are derved ad solved umercally. The results are valdated usg umercal ad theoretcal data avalable the lterature. The aalyss s accomplshed for sphercal shells of dfferet boudary codtos ad radus to thckess ratos. Ths proposed hybrd fte elemet method ca be used effcetly for desg ad aalyss of sphercal shells employed hgh speed arcraft structures.. Itroducto Shells of revoluto, partcularly sphercal shells are oe of the prmary structural elemets hgh speed arcraft. Ther applcatos clude the propellat tak or gas-deployed skrt of space crafts. Free vbrato of sphercal shell has bee vestgated by umerous researchers expermetally ad aalytcally. Kals [,], studyg aalytcally free vbratos shallow sphercal shell, selected used two auxlary varables for the axal ad crcumferetal dsplacemets whle cosderg the effect of logtudal, trasverse ad rotary erta as well as trasverse shear deformato o the o-asymmetrc vbrato of shallow sphercal shells. Navarata [4], Webster [5], Greee et al. [7] used the classcal fte elemet method to study the free vbrato of th sphercal shell. Cohe[3] usg a method of terato lke Stodola s method determed the atural frequeces ad mode shapes of sphercal shell method. Kraus [6] vestgated the case of clamped sphercal shell usg a geeral theory whch cluded the effects of trasverse shear stress ad rotatoal erta. Tessler ad Sprdglozz [8] gave frequeces of 60 clamped sphercal shell ad hemsphercal shell for radus to thckess from 0 to 00 ad ther aalyss was based upo shell theory. Narasmha ad Alwa [9] aalyzed the axsymmetrc free vbrato of clamped sotropc sphercal shell cap. Thck shell aalyss was gve by Gautham ad Gaesa [0] for the aalyss of a 60 clamped ad smply supported sphercal shells, the sem-aalytcal method was used to reduce the dmeso of the problem. The same authors [] vestgated the aalyss of a clamped sotropc hemsphercal shell ( 0 =90. Sa am ad Srdhar Babu [] used the classcal fte elemets method to study the free vbrato of composte sphercal shell cap wth or

6 wthout a cutout. Buchaa ad ch [3] vestgated the case of 60 clamped ad smply supported sphercal shells usg classcal fte elemets method. ecetly, Vetsel et al. [4] used a combed formulato of the boudary elemets method ad fte elemets method to study the free vbrato of a sotropc smply supported hemsphercal shell wth dfferet crcumferetal mode umbers. The objectve of the preset study s to develop a geeral hybrd fte elemet package for predctg the dyamc behavor of sotropc sphercal shells wth boudary codtos whch ca be vared as desred. The soluto scheme s based o the hybrd fte elemet method. Ths method uses dsplacemets fuctos derved from the shell theory stead of polyomals classcal fte elemet method. The elemet s a sphercal frustum stead of the usual tragular or rectagular shell elemet. Ths developed method demostrated precse ad fast covergece wth few elemets. O the other had, the preset theory, because of ts usage of shell classcal theory for the dsplacemet fuctos ca easly be adapted to take the hydrodyamc effects to accout. Fally, aga because of the use of shell classcal theory, we ca obta the hgh as well as the low frequeces wth hgh accuracy.. Fte elemet formulato I ths study the structure s modeled usg hybrd fte elemet method whch s a combato of sphercal shell theory ad classcal fte elemet method. I ths hybrd fte elemet method, the dsplacemet fuctos are foud from exact soluto of sphercal shell theory rather approxmated by polyomal fuctos doe classcal fte elemet method. I the sphercal coordate system(,, show Fg., fve out of the sx equatos of equlbrum derved referece for sphercal shells uder exteral load are wrtte as follows : N N + + ( N N cot + Q = 0 s N N + + N cot + Q = 0 s Q Q + + Q cot ( N + N = 0 s M M + + ( M M cot Q = 0 s M M + + M cot Q = 0 s ( 3

7 X 3 W U U X Fg.. Geometry of sphercal shell Where N, N, N are membrae stress resultats; M, M, M the bedg stress resultats ad Q, Q the shear forces (Fg.. The sxth equato, whch s a detty equato for sphercal shells, s ot preseted here. X d Q M N N M Q N M d N M Fg.. Stress resultats ad stress couples 4

8 Stra ad dsplacemets for three dsplacemets axalu, radal W ad crcumferetal U are related as follows: U + W U ( + U cot + W ε s ε U U ( + U cot ε s κ U W κ κ U W W + U cot cot s s U U W W + U cot + cot s s s { ε} = = ( DsplacemetsU, W ad V the global cartesa coordate system are related to dsplacemetsu, W ad U dcated Fg 3. by: U s cos 0 U W cos s 0 = W V 0 0 U The stress vector{ σ } s expressed as fucto of stra { ε} by { σ} [ P]{ ε} Where [ P ] s the elastcty matrx for a asotropc shell gve by [ P] = (4 P P 0 P4 P5 0 P P 0 P4 P P = P4 P4 0 P44 P45 0 P5 P5 0 P54 P P P66 Upo substtuto of equatos (, (4 ad (5 to equatos (, a system of equlbrum equatos ca be obtaed as a fucto of dsplacemets: 5 (3 (5

9 3 ( j ( j ( j L U, WU,, P = 0 L U, WU,, P = 0 L U, WU,, P = 0 These three lear partal dfferetals operators L, L ad L 3 are gve the Appedx, ad P j are elemets of the elastcty matrx whch, for a sotopc th shell wth thckess h s gve by: (6 [ P] D ν D ν D D ( ν D = K ν K ν K K 0 ( ν K (7 3 Et Et Where D = s the membrae stffess ad K = ν ν ( s the bedg stffess. The elemet s a crcumferetal sphercal frustum show Fg. 3. It has two odal crcles wth four degrees of freedom; axal, radal, crcumferetal ad rotato at each ode. Ths elemet type makes t possble to use th shell equatos easly to fd the exact soluto of dsplacemet fuctos rather tha a approxmato wth polyomal fuctos as doe classcal fte elemet method. For motos assocated wth the th crcumferetal wave umber we may wrte: ( ( ( ( ( ( ( ( ( U, cos 0 0 u u W, 0 cos 0 = w = [ T] w U, 0 0 s u u (8 The trasversal dsplacemet w ( ca be expressed as: Where 3 w ( = w (9 = ( cos ( cos w = AP + BQ (0 6

10 j W dw d U U Fg. 3. Sphercal frustum elemet 7

11 Ad where P ( cos, ( cos Q respectvely of order ad degree. The expresso of the axal dsplacemet u ( s: Where the coeffcet E s gve by: are the assocated Legedre fuctos of the frst ad secod kds 3 dw u ( = E ψ ( d s ( = λ + k( + ν ( ν E = + + ( ( k ( λ ν The auxlary fucto ψ s gve by the expresso: ( = AP ( cos + BQ ( cos ψ 4 4 Fally the crcumferetal dsplacemet u ( ca be expressed as: 3 ( = The degree s obtaed from the expresso (3 dψ u = Ew + s d (4 Where λ s oe the roots of the cubc equato: = + λ 4 (5 Where λ λ + λ = (6 3 3 h h h3 0 h = 4 h h = + + ν 4 ( k( = + ν ( k( (7 Wth k = h The above equato has three roots wth oe root s real ad two other are complex cojugate roots. The Legedre fuctos P, P, Q adq are a real fuctos whereas ( =, 3 are complex fuctos so we ca put: P, P, Q adq 8

12 P = e( P + Im( P P = e( P Im( P 3 Q = e( Q + Im( Q Q = e( Q Im( Q 3 P = e( P + Im( P P = e( P Im( P 3 Q = e( Q + Im( Q e( Im( Q Q = P 3 (8 Settg ( ( ( ( + = c ( + = c + c 3 ( + = c c E = e E = e e E = e + e 3 3 (9 (0 Substtutg equatos (8, (9 ad (0 equatos (9, ( ad (4 we have: ( = ( cot + u e P e c P A + e cote( P e 3cotIm( P + ( e c + e3c3 e( P + ( e 3c ec3 Im( P ( A + A3 + e3cot e( P e cotim( P ( e 3c ec3 e( P + ( e c + e3c3 Im( P ( A + P s A 4 ( cot + e Q + ecq B A 3 + e cote( Q e 3cotIm( Q + ( e c + e3c3 e( Q + ( e 3c ec3 Im( Q ( B + B3 + e cote( Q e cotim( Q ( e c e c e( Q + ( e c + Q B s 4 + ec 3 3 Im( Q ( B B3 9

13 ( w = P A + e( P ( A + A + Im( P ( A A + Q B + e( Q ( B + B + Im( Q ( B B u ( = e P A s e e( P + e 3 Im( P ( A A3 e3 e( P e Im( P ( A A3 s s + + s s + cotp + ( ( + P A eq B s 4 e e( Q e + 3 Im( Q ( B B3 e3 e( Q e Im( Q ( B B3 s s + + s s + cotq + ( ( + Q B 4 I dervg the above relato we used the recursve relatos: ( dp = cot P d + ( ( + P dq = cot Q d + ( ( + Q Usg matrx formulato, the dsplacemet fuctos ca be expressed as follows: (, (, (, ( ( ( U u W = [ T] w = T C U u [ ][ ]{ } (3 The vector { C } s gve by the expresso: T { C} { A A A3 ( A A3 A4 B B B3 ( B B3 B4} The elemets of matrx [ ] are gve the Appedx. = + + (4 I the fte elemet method, the vector C s elmated favor of dsplacemets at elemets odes. At each fte elemet ode, the three dsplacemets (axal, trasversal ad crcumferetal ad the rotato are appled. The dsplacemet of ode are defed by the vector: 0 (

14 dw u w u { δ } T = dx (5 The elemet Fg. 3 wth two odal les ( ad j ad eght degrees of freedom wll have the followg odal dsplacemet vector: Wth j δ dw j j dw j = u w u u w u [ A]{ C} δ = j d d (6 dw = ( cotp + cp A + cote( P c e( P c 3Im( P ( A A3 d cotim( P + c ( 3e( P + c Im( P ( A A3 + cotq + cq B + cot e( Q + c e( Q c 3Im( Q ( B + B3 + cot Im( Q + c 3e( Q + c Im( Q ( B B3 (7 The terms of matrx [ A ] are obtaed from the values of matrx [ ] ad Now, pre-multplyg by [ A] the degree of freedom: T dw dx are gve the appedx. equato (6 oe obtas the matrx of the costat C as a fucto of δ = (8 { C} [ A] δ j Fally, oe substtutes the vector { C} to equato ad obtas the dsplacemet fuctos as follows: The stra vector { } dsplacemet as: { ε} (, (, (, U δ δ W = [ T][ ][ A] = [ N] δ j δ j U (9 ε ca be determed from the dsplacemet fuctos U, U, W ad the deformato [ T] [ 0] [ ] [ ] [ ]{ } [ T] [ 0] [ ] [ ] [ ][ ] δ Q C Q A [ B ] 0 T 0 T δ j δ = = = δ j Where matrx [ Q ] s gve the appedx. Ths relato ca be used to fd the stress vector, equato (4, terms of the odal degrees of freedom vector: (30

15 = δ j P B δ { σ} [ ][ ] (3 Based o the fte elemet formulato, the local stffess ad mass matrces are: T [ ] = [ ] [ ][ ] loc k B P B da A T [ ] = ρ [ ] [ ] loc m h N N da Where ρ the desty ad h s the thckess of shell. A (3 The surface elemet of the shell wall s da = sdd. After tegratg over, the precedg equatos become j k A Q P Q d A A G A T T T [ ] = π [ ] [ ][ ] s = [ ] loc j m h A d A h A S A T T T [ ] = ρ π [ ] [ ] s = ρ [ ] loc I the global system the elemet stffess ad mass matrces are T T [ k] [ LG] A [ G] A [ LG] = T T [ m] ρh[ LG] A [ S] A [ LG] (33 = (34 Where [ LG] s cos cos s = sj cosj cosj sj (35 From these equatos, oe ca assemble the mass ad stffess matrx for each elemet to obta the mass ad stffess matrces for the whole shell: [ M ] ad[ K ]. Each elemetary matrx s 8x8, therefore the fal dmesos of [ M ] ad [ K ] wll be 4(N+ where N s the umber of elemets of the shell.

16 3. Numercal results I order to test the effcecy ad the accuracy of our model, we used the theory developed ths paper to calculate the atural frequeces ad modes of uform th elastc sphercal shell, whch were both o-shallow( 0 <30 ad shallow, of varous dmesos ad uder dfferet boudary codtos. These cases have already bee vestgated by other authors usg others methods. For purposes of comparsos amog the atural frequeces obtaed, t s emetly practcal at ths stage to troduce the o-dmesoal atural frequecy: Where: ω s the atural agular frequecy. s the radus of the referece surface. ρ s the desty. E s the modulus of elastcty. ρ Ω= ω E (36 0 Fg.4. Defto of agle 0 3

17 3. Case : clamped sphercal shell wth 0 =0 Narassha ad Alwar [9] vestgated the case of a axsymmetrc clamped sphercal shell. The aalyss s based o the applcato of the Chebyshev-Galerk spectral method for the evaluato of free vbrato frequeces ad mode shapes. Sa am ad Sreedhar Babu [] aalyzed the same case wth the classcal fte elemet method usg 80 elemets. Each elemet s a eght oded degeerated soparametrc shell elemet wth e degrees of freedom at each ode. Wth our model ad usg 6 fte elemets, the atural were computed, the results are show table. Mode Preset Sa am ad Sreedhar babu[] Narassha ad Alwar [9] Table : Normalzed atural frequeces for 0 clamped sphercal shell wth 00 h = 3. Case : clamped sphercal shell wth 0 =30 Ths case was vestgated aalytcally by Kals [] usg classcal theory ad trasverse vbrato theory. Wth our theory, we used 8 fte elemets to study the sphercal shell wth the results show table. The frequeces we obtaed wth our model are very comparable to Kal s values. The maxm dsplacemet values were: W max = U max ( 3.6 max = U max It was observed that at lowest atural frequecy, moto of the sphercal shell s maly domated by ts radal compoet. W (.54 Mode Preset theory Kals [] Table : Normalzed atural frequeces for 30 clamped sphercal shell wth 0 h = 4

18 3.3 Case 3: sphercal shell 0 =60 uder the three boudary codtos: clamped, smply upported ad free The free axsymmetrc vbrato of the sphercal shell ths case was studed by Kals [], Cohe [3], Navarata [4], Webster [5], Greee et al [7], Tessler ad Sprdglozz [8], Gautham ad Gaesa [0] ad Buchaa ad ch [3]. I the preset vestgato, the shell was vestgated wth 0 elemets; the results are gve respectvely for clamped, smply supported ad free shells tables 3, 4 ad 5. The atural modes correspodg to the lowest shell atural frequeces uder the two boudary codtos are llustrated fgures 5 ad 6. They reveal that at the lowest atural frequecy, sphercal shell moto s radal. It s easy to see that all dsplacemetsu, W ad U are all zero at the top ( = 0 of the sphercal shell. Mode Kals[] Navarata [4] Webster [5] Tessler ad Sprdglozz [8] Gautham ad Gaesa [ 0] Buchaa ad ch [3] Preset theory Table 3: Normalzed atural frequeces for 60 clamped sphercal shell wth 0 h = 5

19 Mode Kals [] Navarata [4] Greee et al [7] Cohe Table 4: Normalzed atural frequeces for 60 smply supported sphercal shell wth 0 h = [3] Gautham ad Gaesa [ 0] Buchaa ad ch [3] Preset Theory Fg.5. Dsplacemets versus coordate for clamped sphercal shell 0 =60 6

20 Mode Kals [] Navarata [4] Webster [5] Preset theory Table 5: Normalzed atural frequeces for 60 free sphercal shell wth 0 h = Fg.5. Dsplacemets versus coordate for smply supported sphercal shell 0 =60 7

21 3.4 Case 4: Sphercal shell wth 0 =90 Kraus [6] vestgated the case of smply supported sphercal shell usg a geeral theory whch cluded the effects of trasverse shear stress ad rotatoal erta. For cases both wth ad wthout these effects, he determed the atural frequeces for the shell moto that was depedet of for crcumferetal mode umber = 0. Tessler ad Sprdglozz [8], Gautham ad Gaesa [] aalyzed the case of clamped hemsphercal shell. Vetsel et al. [4] studed the case of smply supported sphercal shell usg the boudary elemets method for varous crcumferetal mode umbers( = 0, =, =. Wth our model ad usg fte elemets, the atural frequeces were computed for clamped ad smply supported shells. The results are show respectvely table 7 ad table 8. The maxmum dsplacemets values are: ( U max = W max = 0.37 ( U max = The result s that at the lowest atural frequecy, the moto of sphercal shell s predomately by the axal dsplacemet. Mode Tessler ad Sprdglozz [8] Gautham ad Gaesa [] Preset theory Table 7 : Normalzed atural frequeces for 90 clamped sphercal shell wth 0 h = 8

22 Mode Kraus [6] 0 h = Kraus [6] h = Table 8: Normalzed atural frequeces for 90 smply supported sphercal shell 50 Vetsel et al.[4] h = 00 Preset theory h = Cocluso The purpose of the vestgato descrbed ths paper s to determe the atural frequeces ad shape modes of free vbratos of sphercal shell. The modal s based o hybrd approach combg the classcal fte elemet method ad the classcal shell theory. Ths theoretcal approach s much more precse tha usual fte elemet methods because the dsplacemet fuctos are derved from exact solutos of equlbrum equatos for sphercal shells. The mass ad stffess matrces are determed by umercal tegrato. The results obtaed for sphercal shells wth dfferet agles ad dfferet boudary codtos are compared wth results avalable the lterature. Very good agreemet was foud. Ths approach resulted a very precse elemet that leads to fast covergece ad less umercal dffcultes from the computatoal pot of vew. Because of ts use of classcal shell theory for the dsplacemet fuctos, the preseted method may easly be adapted to take flud-structure effects to accout. A paper uder preparato o the effect flud o vbratos of shells cofrms ths approach. For the same reaso, we ca obta the hgh as well as low frequeces wth very good accuracy. 9

23 Appedx P P4 U U L ( U, U, W = + W W cot( P P4 U U + + U cot( W U cot( W cot( s( s( P U 4 P44 W U W cot( P5 P45 U W W U W W + U cot( cot( U cot( cot( cot( s( s ( s( s ( P P U W cot( P P U U cot( W cot( s( P P U W cot( P5 P55 U W W U cot( cot( cot( + + s( s ( P33 P63 U Uϕ U cot( s( s( P36 P66 U U ϕ cot( W W + U cot( s( s( s( s( P P5 U L ( U, U, W = + W + s( P P5 U + + cot( + U + W s( s( P4 P54 U W + + s( P5 P55 U W W U + cot( cot( + + s( s( s ( P33 P63 U Uϕ U Uϕ U cot( + + U cot( cot( s( s( P36 P66 U Uϕ cot( W W U U ϕ cot( W W + U cot( U cot( cot t( s( s( s( s( s( s( 0

24 L3 ( U, U, W = + + W P5 P5 U W W U + cot( cot( + s( s ( 4 5 P P U P P U U cot( W s( P U 4 P4 W + P P + U U cot( + W + W U U + P4 s( + W + P5 + W s( P4 P 5 U U + cot( U cot( W U cot( W s( s( U U + P4 s( + U cot( + W + P5 + U cot( + W s( s( s( U U P44 P 54 W W + cot( P44 s( U W U W s( + P 54 P45 P 55 U W W U W W + cot( U cot( cot( U cot( cot( s( s ( s( s ( + P 3 s( U W W U W W s( + cot( cot( + + cot( cot( s( s ( s( s ( 45 U P 55 U U P 63 U ϕ U ϕ + + U cot( 3cot ( + + U cot( s( s( s( P U U cot( W W U U cot( W W s( s( s( s( s( s( s( 66 ϕ ϕ + U cot( 3cot ( + U cot( + U

25 ( (, = ecot P + ecp (, = e cote( P e cotim( P + ( e c + e c e( P + ( e c e c Im( P (,3 = e cote( P e cotim( P ( e c e c e( P + ( e c + e c Im( P (, 4 = P s ( (, 5 = ecot Q + ecq (,6 = e cote( Q e cotim( Q + ( e c + e c e( Q + ( e c e c Im( Q (,7 = e cote( Q e cotim( Q ( e c e c e( Q + ( e c + e c Im( Q (,8 = Q s (, = P (, = e( P ( =,3 Im( P (, 4 = 0 (,5 = Q (,6 = e( Q ( =,7 Im( Q (,8 = ( 3, = e P s (3, = e e( P e 3 Im( P s s (3,3 = e3 e( P e Im( P s s (3, 4 = cotp + ( ( + P ( 3,5 = e Q s

26 (3,6 = e e( Q e 3 Im( Q s s (3,7 = e3 e( Q e Im( Q s s (3,8 = cot Q + ( ( + Q dw A(, j = (, j, A(, j = (, j, A(3, j = ( j, A(4, j = (3, j wth = A(5, j = (, j, d dw A(6, j = (, j ; A(7, j = ( j, A(8, j = (3, j wth = j d j=,,8 3

27 e Q(, = e c e ( cot P c cotp s r Q ( e c e c e P s (, = ( + cot + e( + ( e3c ec3 e3( cot Im( P + + s ( ec + ec 3 3 cote( P ( ec 3 ec 3 cotim( P Q(,3 = ( e3c ec3 e3( cot e( P + + s + ( ec e3c3 e( cot Im( P s + ( ec 3 ec 3 cote( P ( ec + ec 3 3 cotim( P Q(, 4 = ( + cot P ( ( + P s s e Q(,5 = e c e ( cot Q c cotq s r Q ( e c e c e Q s (,6 = ( + cot + e( + ( e3c ec3 e3( cot Im( Q + + s ( ec + ec 3 3 cot e( Q ( ec 3 ec 3 cot Im( Q Q(,7 = ( e3c ec3 e3( cot e( Q + + s + ( ec e3c3 e( cot Im( Q s + ( ec 3 ec 3 cote( Q ( ec + ec 3 3 cotim( Q Q(,8 = ( + cot Q ( ( + Q s s 4

28 e Q(, = e ( cot P c cotp + + s r e3 Q(, = e ( cot e( P ( cot Im( P + s + s + ( ec + ec 3 3cote( P + ( ec 3 ec 3cotIm( P e3 Q(,3 = ( + cot e( P + e ( cot Im( P s + s ( ec 3 ec 3cote( P + ( ec + ec 3 3cotIm( P Q(, 4 = ( + cot P + ( ( + P s s e Q(,5 = e ( cot Q + + s c cotq r e3 Q(,6 = e ( cot e( Q ( cot Im( Q + s + s + ( ec + ec 3 3cote( Q + ( ec 3 ec 3cotIm( Q e3 Q(,7 = ( + cot e( Q e + ( + cot s s Im( Q ( ec 3 ec 3cote( Q + ( ec + ec 3 3cotIm( Q Q(,8 = ( + cot Q + ( ( + Q s s 5

29 Q(3, = e( + cot P ec P s s Q(3, = e( + cote( P + e 3( + cotim( P s s ( ec + ec 3 3 e( P ( ec 3 ec 3 Im( P s s Q(3, 3 = e3 ( + cot e( P + e ( + cot Im( P s s + ( ec 3 ec 3 e( P ( ec + ec 3 3 Im( P s s Q(3, 4 = ( + ( + + cot P ( ( + cotp s Q(3, 5 = e( + cot Q c Q s s Q(3,6 = e( + cote( Q + e 3( + cotim( Q s s ( ec + ec 3 3 e( Q ( ec 3 ec 3 Im( Q s s Q(3,7 = e3( + cote( Q + e ( + cotim( Q s s + ( ec 3 e c3 e( Q ( ec + ec 3 3 Im( Q s s Q(3,8 = ( + ( + + cot Q ( ( + cotq s 6

30 e Q(4, = ( e c + ( e + cot P c cotp s Q(4, = e c ec e cot e P s ( 3 3 ( ( + e3c e c3+ e3 + P s + Q(4,3 = e3c e c3+ e3 + cot e P s ( cot Im ( ( e c ec 3 3 cote( P ec ( ( 3 e c3 cot Im P ( ( s + + Q(4, 4 = ( + cot P ( ( + P s s ( e c ec 3 3 ( e cot Im ( P ec 3 ( e c3 cot e( P ( e c ec 3 3 cot Im ( P e Q(4,5 = ( e c + ( e + cot Q c cotq s Q(4, 6 = e c ec e cot e Q s ( 3 3 ( ( + ec 3 ( e c3+ e3 + cot Im ( Q s + Q(4, 7 = e 3c ( e c3+ e3 + cot e ( Q s ( e c ec 3 3 cote( Q ec 3 ( e c3 cotim ( Q + ( e c+ ec 3 3+ ( e + cot Im ( Q s + ec 3 e c3 cot e Q e c ec 3 3 cot Im Q + Q(4,8 = ( + cot Q ( ( + Q s s ( ( ( ( 7

31 ( e Q(5, = ( e cot + P + c cotp s e Q(5, = e cot e cot Im + P P + s s e Q(5,3 s s ec 3 e c3 cot e P e c ec 3 3 cot Im P + + Q(5, 4 = ( + cotp + ( ( + P s s ( ( 3 ( ( e c ec 3 3 cote( P ec 3 ( e c3 cotim ( P 3 = cot + e( P + ( e ( cot + Im P ( ( ( ( e Q(5,5 = ( e cot + Q + c cotq s ( e Q(5, 6 = e cot + e cot Im Q + Q s s e3 Q(5, 7 = cot e + ( Q + e cot + Im Q s s ec 3 e c3 cot e Q e c ec 3 3 cot Im Q + + Q(5,8 = ( + cotq + ( ( + Q s s ( ( 3 ( ( e c ec 3 3 cote( Q ec 3 ( e c3 cotim ( Q ( ( ( ( ( ( 8

32 Q(6, = ( + ( e cot P + ( e c P s s ( + ( + Q(6, = e cot e P + e cot Im P s s Q(6,3 ( ( 3 ( + s s ( e c ec 3 3 e( P ec 3 ( e c3 Im ( P ( + ( + s s = e 3 cote( P + ( e ( cotim P + ec e c P e c ec P + s s ( e( ( Im ( Q(6, 4 = ( + ( + + cot P ( ( + cotp s Q(6,5 = ( + ( e cotq + ( e c Q s s ( + ( + Q(6, 6 = e cot e Q + e cot Im Q s s ( ( 3 ( + s s ( e c ec 3 3 e( Q ec 3 ( e c3 Im ( Q ( + ( + Q(6, 7 e cot e Q s s = 3 ( + ( e cot Im ( Q + ec e c Q e c ec Q + s s ( e( ( Im ( Q(6,8 = ( + ( + + cot Q ( ( + cotq s I dervg the above relato we used the recursve relatos: d P d = ( ( cot P cot ( ( + P s dq d = ( ( cot Q cot ( ( + Q s 9

33 efereces [] Kals A., Free o-symmetrc vbratos of shallow sphercal shells, Joural of the Acoustcal Socety of Amerca 33 ( [] Kals A., Effect of bedg o vbrato of sphercal shell, Joural of the Acoustcal Socety of Amerca 36 ( [3] Cohe G.A., Computer aalyss of asymmetrc free vbratos of rg stffeed orthotropc shells of revoluto, AIAA Joural 3 ( [4] Navarata D.., Natural Vbrato of Deep Sphercal Shells, AIAA Joural 4 ( [5] Webster J.J., Free vbratos of shells of revoluto usg rg fte elemets, Iteratoal Joural of Mechacal Sceces 9( [6] Kraus H., Th elastc shells, Joh Wley ad Sos, New York, 967. [7] Greee B.E., Joes.E., Mc Lay.W. ad Strome D.., Dyamc aalyss of shells usg doubly curved fte elemets, Proceedgs of The Secod Coferece o Matrx methods Structural Mechacs, T-68-50( [8] Tessler A., Sprchglozz L., esolvg membrae ad shear lockg pheomea curved deformable axsymmetrc shell elemet, Iteratoal Joural for Numercal Methods Egeerg, 6( [9] Narasmha M.C., Alwar.S., Free vbrato aalyss of lamated orthotropc sphercal shell, Joural of Soud ad Vbrato, 54( [0] Gautham B. P., Gaesa N., Free vbrato aalyss of thck sphercal shells, Computers ad Structures Joural, 45( [] Gautham B. P., Gaesa N., Free vbrato characterstcs of sotropc ad lamated orthotropc shell caps, Joural of Soud ad Vbrato, 04( [] Sa am K.S., Sreedhar Babu T., Free vbrato of composte sphercal shell cap wth ad wthout a cutout, Computers ad Structures Joural, 80( [3] Buchaa G.., ch B.S., Effect of boudary codtos o free vbratos of thck sotropc sphercal shells, Joural of Vbrato ad Cotrol 8( [4] Vetsel E.S, Naumeko V., Strelkova E., Yeseleva E., Free vbratos of shells of revoluto flled wth flud, Joural of Egeerg Aalyss wth Boudary Elemets 34(

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